ABCF->ab-angle angle

Percentage Accurate: 53.1% → 81.3%

Time: 7.5s

Alternatives: 15

Speedup: 2.6×

Specification

Math

\[\begin{array}{l} \\ 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (*
  180.0
  (/
   (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
   (PI))))

Initial Program: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (*
  180.0
  (/
   (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
   (PI))))

Alternative 1: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;A \leq -5.8 \cdot 10^{+134}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= A -5.8e+134)
     (* t_0 (atan (* (/ B A) 0.5)))
     (* (atan (/ (- (- C A) (hypot (- A C) B)) B)) t_0))))

Derivation

1. Split input into 2 regimes

2. if A < -5.80000000000000023e134

   1. Initial program 7.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 82.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 82.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 82.6

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -5.80000000000000023e134 < A

   1. Initial program 64.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 64.0

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 85.1%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right)\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(t\_1 + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (atan
          (* (- (- C A) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))) (/ 1.0 B))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (* (/ (atan (- t_1 1.0)) (PI)) 180.0)
     (if (<= t_0 0.0)
       (* (/ 180.0 (PI)) (atan (* (/ B A) 0.5)))
       (* (/ (atan (+ t_1 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < -0.5

   1. Initial program 60.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. associate--r+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)}}{\mathsf{PI}\left(\right)} \]

      3. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      4. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 70.8

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 70.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right)} \]

   if -0.5 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))) < -0.0

   1. Initial program 18.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 51.0

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 51.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 51.2

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 51.2%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -0.0 < (atan.f64 (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))

   1. Initial program 56.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 71.7

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 71.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right) \leq -0.5:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} - 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;\tan^{-1} \left(\left(\left(C - A\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \frac{1}{B}\right) \leq 0:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.2e+131)
   (* (/ 180.0 (PI)) (atan (* (/ B A) 0.5)))
   (if (<= A 2.4e+42)
     (/ (atan (/ (- C (hypot B C)) B)) (* 0.005555555555555556 (PI)))
     (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -5.2e131

   1. Initial program 7.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 82.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 82.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 82.6

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -5.2e131 < A < 2.3999999999999999e42

   1. Initial program 57.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 57.1

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 80.5

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower-hypot.f64 78.2

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 78.2%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if 2.3999999999999999e42 < A

   1. Initial program 83.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 89.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;A \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= A -5.2e+131)
     (* t_0 (atan (* (/ B A) 0.5)))
     (if (<= A 2.4e+42)
       (* (atan (/ (- C (hypot B C)) B)) t_0)
       (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if A < -5.2e131

   1. Initial program 7.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 82.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 82.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 82.6

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -5.2e131 < A < 2.3999999999999999e42

   1. Initial program 57.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 57.1

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right) \]

      2. unpow2 N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right) \]

      3. unpow2 N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right) \]

      4. lower-hypot.f64 78.2

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right) \]

   7. Applied rewrites 78.2%

      \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right) \]

   if 2.3999999999999999e42 < A

   1. Initial program 83.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 89.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(B, C\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -5.2e+131)
   (* (/ 180.0 (PI)) (atan (* (/ B A) 0.5)))
   (if (<= A 2.4e+42)
     (* (/ (atan (/ (- C (hypot C B)) B)) (PI)) 180.0)
     (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -5.2e131

   1. Initial program 7.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 82.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 82.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 82.6

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -5.2e131 < A < 2.3999999999999999e42

   1. Initial program 57.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \sqrt{{B}^{2} + {C}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \sqrt{{B}^{2} + {C}^{2}}}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower--.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{C \cdot C} + {B}^{2}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      5. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \sqrt{C \cdot C + \color{blue}{B \cdot B}}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-hypot.f64 78.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(C, B\right)}}{B}\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 78.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}}{\mathsf{PI}\left(\right)} \]

   if 2.3999999999999999e42 < A

   1. Initial program 83.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 89.4

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 89.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - \mathsf{hypot}\left(C, B\right)}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;A \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.16 \cdot 10^{-44}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}}{B}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (/ 180.0 (PI))))
   (if (<= A -3.4e-11)
     (* t_0 (atan (* (/ B A) 0.5)))
     (if (<= A 2.16e-44)
       (* (atan (/ (- (- C A) (sqrt (fma C C (* B B)))) B)) t_0)
       (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0)))))

Derivation

1. Split input into 3 regimes

2. if A < -3.3999999999999999e-11

   1. Initial program 18.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 70.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 70.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 70.7

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -3.3999999999999999e-11 < A < 2.15999999999999988e-44

   1. Initial program 58.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\color{blue}{{B}^{2} + {C}^{2}}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\color{blue}{\mathsf{fma}\left(C, C, {B}^{2}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, \color{blue}{B \cdot B}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, \color{blue}{B \cdot B}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 58.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\color{blue}{\mathsf{fma}\left(C, C, B \cdot B\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. div-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{180}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \frac{1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

   7. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if 2.15999999999999988e-44 < A

   1. Initial program 81.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 88.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 88.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 2.16 \cdot 10^{-44}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -2.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -2.9e-11)
   (* (/ 180.0 (PI)) (atan (* (/ B A) 0.5)))
   (if (<= A -1.6e-104)
     (/ (atan (/ (- C B) B)) (* 0.005555555555555556 (PI)))
     (* (/ (atan (+ (/ (- C A) B) 1.0)) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -2.9e-11

   1. Initial program 18.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 70.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 70.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 70.7

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -2.9e-11 < A < -1.59999999999999994e-104

   1. Initial program 55.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 55.4

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 76.6

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower-hypot.f64 76.7

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 76.7%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 61.7%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if -1.59999999999999994e-104 < A

   1. Initial program 68.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \left(\frac{C}{B} - \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. div-sub N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(1 + \color{blue}{\frac{C - A}{B}}\right)}{\mathsf{PI}\left(\right)} \]

      3. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} + 1\right)}{\mathsf{PI}\left(\right)} \]

      6. lower--.f64 66.9

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} + 1\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 66.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + 1\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - A}{B} + 1\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -2.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-2 \cdot \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -2.9e-11)
   (* (/ 180.0 (PI)) (atan (* (/ B A) 0.5)))
   (if (<= A 9e-39)
     (/ (atan (/ (- C B) B)) (* 0.005555555555555556 (PI)))
     (* (/ (atan (* -2.0 (/ A B))) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -2.9e-11

   1. Initial program 18.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 70.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 70.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. associate-/r/ N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      8. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{B}{A} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      9. lower-*.f64 70.7

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   7. Applied rewrites 70.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{B}{A}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

   if -2.9e-11 < A < 9.0000000000000002e-39

   1. Initial program 58.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 58.5

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 82.4

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower-hypot.f64 80.1

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 80.1%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 48.2%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if 9.0000000000000002e-39 < A

   1. Initial program 82.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-2 \cdot \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A}{B} \cdot -2\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A}{B} \cdot -2\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 71.1

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{A}{B}} \cdot -2\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 71.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A}{B} \cdot -2\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-2 \cdot \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;A \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-2 \cdot \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= A -2.55e-11)
   (* (/ (atan (* (/ B A) 0.5)) (PI)) 180.0)
   (if (<= A 9e-39)
     (/ (atan (/ (- C B) B)) (* 0.005555555555555556 (PI)))
     (* (/ (atan (* -2.0 (/ A B))) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if A < -2.54999999999999992e-11

   1. Initial program 18.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{B}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot \frac{1}{2}\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 70.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{A}} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 70.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{B}{A} \cdot 0.5\right)}}{\mathsf{PI}\left(\right)} \]

   if -2.54999999999999992e-11 < A < 9.0000000000000002e-39

   1. Initial program 58.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 58.5

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 82.4

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower-hypot.f64 80.1

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 80.1%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 48.2%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if 9.0000000000000002e-39 < A

   1. Initial program 82.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in A around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-2 \cdot \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A}{B} \cdot -2\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A}{B} \cdot -2\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 71.1

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{A}{B}} \cdot -2\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 71.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{A}{B} \cdot -2\right)}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{B}{A} \cdot 0.5\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-39}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(-2 \cdot \frac{A}{B}\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;B \leq -4.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-244}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (PI))))
   (if (<= B -4.3e-147)
     (* (/ (atan 1.0) (PI)) 180.0)
     (if (<= B 4e-244)
       (/ (atan (/ (- A) B)) t_0)
       (/ (atan (/ (- C B) B)) t_0)))))

Derivation

1. Split input into 3 regimes

2. if B < -4.3000000000000001e-147

   1. Initial program 49.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{1}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

   if -4.3000000000000001e-147 < B < 3.9999999999999997e-244

   1. Initial program 57.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 57.3

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 83.2

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 83.2%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. associate--r+ N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. div-sub N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      6. lower--.f64 51.3

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 51.3%

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in A around inf

       \[\leadsto \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\frac{A}{B}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 41.0%

       \[\leadsto \frac{\tan^{-1} \left(\frac{-A}{\color{blue}{B}}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if 3.9999999999999997e-244 < B

   1. Initial program 55.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 55.0

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 78.4

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower-hypot.f64 65.1

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 65.1%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.4%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-244}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -4.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -4.3e-147)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 2.8e-223)
     (/ (atan (/ (- A) B)) (* 0.005555555555555556 (PI)))
     (if (<= B 6.5e-43)
       (* (/ (atan 0.0) (PI)) 180.0)
       (* (/ (atan -1.0) (PI)) 180.0)))))

Derivation

1. Split input into 4 regimes

2. if B < -4.3000000000000001e-147

   1. Initial program 49.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{1}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

   if -4.3000000000000001e-147 < B < 2.80000000000000015e-223

   1. Initial program 60.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 60.2

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 84.4

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in B around inf

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C}{B} - \color{blue}{\left(\frac{A}{B} + 1\right)}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. associate--r+ N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\left(\frac{C}{B} - \frac{A}{B}\right) - 1\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. div-sub N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\color{blue}{\frac{C - A}{B}} - 1\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      6. lower--.f64 54.6

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - A}}{B} - 1\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 54.6%

      \[\leadsto \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} - 1\right)}}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in A around inf

       \[\leadsto \frac{\tan^{-1} \left(-1 \cdot \color{blue}{\frac{A}{B}}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 43.4%

       \[\leadsto \frac{\tan^{-1} \left(\frac{-A}{\color{blue}{B}}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if 2.80000000000000015e-223 < B < 6.50000000000000001e-43

   1. Initial program 46.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      2. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. mul0-lft N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. div0 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{0}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval 39.2

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{0}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 39.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{0}}{\mathsf{PI}\left(\right)} \]

   if 6.50000000000000001e-43 < B

   1. Initial program 56.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-A}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;C \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-0.5}{C} \cdot B\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= C 1.7e-68)
   (/ (atan (/ (- C B) B)) (* 0.005555555555555556 (PI)))
   (* (/ (atan (* (/ -0.5 C) B)) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if C < 1.70000000000000009e-68

   1. Initial program 65.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{180}{1}} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{180}{1} \cdot \color{blue}{\frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1 \cdot \mathsf{PI}\left(\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot 1}} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{1}} \]

      7. /-rgt-identity N/A

         \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      9. lower-/.f64 65.6

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \cdot \tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \color{blue}{\left(\left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \frac{1}{B}\right)} \]

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{180}{\mathsf{PI}\left(\right)} \cdot \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \frac{180}{\mathsf{PI}\left(\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{180}{\mathsf{PI}\left(\right)}} \]

      4. clear-num N/A

         \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(A - C, B\right)}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}}} \]

      7. lift-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\sqrt{\left(A - C\right) \cdot \left(A - C\right) + B \cdot B}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      8. +-commutative N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{\color{blue}{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      9. lower-hypot.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}}{B}\right)}{\frac{\mathsf{PI}\left(\right)}{180}} \]

      10. div-inv N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}}} \]

      12. metadata-eval 86.9

          \[\leadsto \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{0.005555555555555556}} \]

   6. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556}} \]

   7. Taylor expanded in A around 0

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \sqrt{{B}^{2} + {C}^{2}}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]

      4. lower-hypot.f64 72.7

         \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{\mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   9. Applied rewrites 72.7%

      \[\leadsto \frac{\tan^{-1} \left(\frac{\color{blue}{C - \mathsf{hypot}\left(B, C\right)}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot \frac{1}{180}} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.4%

       \[\leadsto \frac{\tan^{-1} \left(\frac{C - \color{blue}{B}}{B}\right)}{\mathsf{PI}\left(\right) \cdot 0.005555555555555556} \]

   if 1.70000000000000009e-68 < C

   1. Initial program 29.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{B}{C} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]

      2. *-commutative N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{B}{C} \cdot \frac{-1}{2}} + -1 \cdot \frac{A + -1 \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      5. mul0-lft N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      6. div0 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + -1 \cdot \color{blue}{0}\right)}{\mathsf{PI}\left(\right)} \]

      7. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{B}{C} \cdot \frac{-1}{2} + \color{blue}{0}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, \frac{-1}{2}, 0\right)\right)}}{\mathsf{PI}\left(\right)} \]

      9. lower-/.f64 59.6

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{fma}\left(\color{blue}{\frac{B}{C}}, -0.5, 0\right)\right)}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 59.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{B}{C}, -0.5, 0\right)\right)}}{\mathsf{PI}\left(\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(B \cdot \color{blue}{\frac{-0.5}{C}}\right)}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{C - B}{B}\right)}{0.005555555555555556 \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{-0.5}{C} \cdot B\right)}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B -3.2e-112)
   (* (/ (atan 1.0) (PI)) 180.0)
   (if (<= B 6.5e-43)
     (* (/ (atan 0.0) (PI)) 180.0)
     (* (/ (atan -1.0) (PI)) 180.0))))

Derivation

1. Split input into 3 regimes

2. if B < -3.19999999999999993e-112

   1. Initial program 51.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{1}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{1}}{\mathsf{PI}\left(\right)} \]

   if -3.19999999999999993e-112 < B < 6.50000000000000001e-43

   1. Initial program 53.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      2. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. mul0-lft N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. div0 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{0}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval 34.0

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{0}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 34.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{0}}{\mathsf{PI}\left(\right)} \]

   if 6.50000000000000001e-43 < B

   1. Initial program 56.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\tan^{-1} 1}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (if (<= B 6.5e-43)
   (* (/ (atan 0.0) (PI)) 180.0)
   (* (/ (atan -1.0) (PI)) 180.0)))

Derivation

1. Split input into 2 regimes

2. if B < 6.50000000000000001e-43

   1. Initial program 52.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{\left(-1 + 1\right) \cdot A}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      2. metadata-eval N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{0} \cdot A}{B}\right)}{\mathsf{PI}\left(\right)} \]

      3. mul0-lft N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      4. div0 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-1 \cdot \color{blue}{0}\right)}{\mathsf{PI}\left(\right)} \]

      5. metadata-eval 21.5

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{0}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 21.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{0}}{\mathsf{PI}\left(\right)} \]

   if 6.50000000000000001e-43 < B

   1. Initial program 56.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\tan^{-1} 0}{\mathsf{PI}\left(\right)} \cdot 180\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 20.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180 \end{array} \]

FPCore

(FPCore (A B C) :precision binary64 (* (/ (atan -1.0) (PI)) 180.0))

Derivation

1. Initial program 53.8%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

5. Applied rewrites 21.3%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{-1}}{\mathsf{PI}\left(\right)} \]

6. Final simplification 21.3%

   \[\leadsto \frac{\tan^{-1} -1}{\mathsf{PI}\left(\right)} \cdot 180 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (A B C)
  :name "ABCF->ab-angle angle"
  :precision binary64
  (* 180.0 (/ (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))) (PI))))